Real Hardy space H1

Summary

The real Hardy space $H^1(\R^n)$ is the space of integrable functions $f$ on $\R^n$ whose Riesz transforms $R_1 f, \dots, R_n f$ are also integrable. It is a proper subspace of $L^1(\R^n)$, and its elements have integral zero. It is the right replacement for $L^1$ in harmonic analysis. Singular integrals such as the Hilbert and Riesz transforms map $H^1$ to $L^1$, but not $L^1$ to $L^1$. Its dual space is the space BMO of functions of bounded mean oscillation, and its elements are sums of simple building blocks called atoms. For $n = 1$ it is the space of real parts of boundary values of functions in the holomorphic Hardy space $H^1$ of the upper half-plane.